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If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The complement of the Cantor ternary set is an example of a fractal string. In arithmetical terms, the Cantor set consists of all real numbers of the unit interval [ 0 , 1 ] {\displaystyle [0,1]} that do not require the digit 1 in order to be expressed as a ternary (base 3) fraction.
The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty ).
A coanalytic set is the complement of an analytic set cofinal A subset of a poset is called cofinal if every element of the poset is at most some element of the subset. cof cofinality cofinality 1. The cofinality of a poset (especially an ordinal or cardinal) is the smallest cardinality of a cofinal subset 2.
For a finite-dimensional inner product space of dimension , the orthogonal complement of a -dimensional subspace is an ()-dimensional subspace, and the double orthogonal complement is the original subspace: =.
But LaTeX command \complement is certainly not intended to be used in exponent (otherwise, its use should be \complement{A}). I have often encountered the notation ∁ A , {\displaystyle \complement A,} which has been introduced by Bourbaki (see [1] for an early use, and [2] for a previous Bourbaki's text that uses a different notation).
Set complement: A ′ is the complement of the set A (other notation also exists). [9] The negation of an event in probability theory: Pr(A ′) = 1 − Pr(A) (other notation also exists). The result of a transformation: Tx = x ′ The transpose of a matrix (other notation also exists) The dual of a vector space